Voltage Ratio and Transformer Polarity

Returning to the discussion of the ideal transformer with N₁ primary turns and N₂ secondary turns, let the instantaneous voltage applied to the primary winding be v₁ v, and let the polarity be as shown in Fig. 6-4. The time-varying flux in the core of the transformer induces an instantaneous emf e₁ in the primary winding, and, since the windings of an ideal transformer are assumed to be without resistance, e₁ must equal v₁. Then

[6-10]

The sense of the induced emf e₁ in Eq. 6-10 and in Fig. 6-4 is opposite that in Eq. 6-9, accounting for the omission of the minus sign in Eq. 6-10.

In the ideal transformer there is no leakage flux. The flux everywhere in the core has the same value and is changing at the same rate at a given instant. Thus, all of the flux links both windings of the ideal transformer and is considered as the mutual flux Φ. Hence, the voltage induced in the secondary winding is

[6-11]

and has the polarity shown in Fig. 6-4. Since the secondary winding of the ideal transformer is assumed to have no resistance, the secondary terminal voltage v₂ must equal the induced secondary voltage e₂ even when the transformer carries load. Hence

[6-12]

The ratio of primary to secondary turns N₁/N₂ of a transformer is called the turns ratio. Division of Eq. 6-10 by Eq. 6-12 shows that, in an ideal transformer, the voltage ratio equals the turns ratio, thus

[6-13]

Also, if V₁, E₁, V₂, and E₂ are the corresponding primary and secondary voltage phasors, we have

[6-14]

It is important to note that the induced emfs e₁ and e₂ as shown in Fig. 6-4 have like polarities. Therefore, the induced voltage phasors E₁ and E₂ are in phase with each other and not 180° out of phase. In the ideal transformer, the terminal voltage phasors V₁ and V₂ are also in phase with each other for the convention adopted in Fig. 6-4. If, in Fig. 6-4, the plus sign of either e₁ or e₂ - but one of them only, were placed at the bottom of the winding, then the induced emf phasors E₁ and E₂ would be 180° out of phase with each other. The black dots near the tops of the two windings are called polarity marks. These indicate that the upper terminals, or the marked terminals, have like polarities, just as the unmarked terminals have like polarities - but opposite those of the marked terminals. One way of establishing the significance of polarity marks is by considering the magnetic effect on the core when positive currents enter both windings either at the marked terminals or when positive currents enter both windings at the unmarked terminals. In either case, the mmfs of the two currents in the two windings add.